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[parent] opposite number (Definition)

The opposite number of a number $ a$ is such a number $ -a$ that

$\displaystyle -a\!+\!a = 0.$

Some properties:

  • $ -0 = 0$
  • $ -(-a) = a$
  • $ -(a\!+\!b) = (-a)\!+\!(-b)$
  • $ -(a\!\cdot\!b) = a\!\cdot\!(-b) = (-a)\!\cdot\!b$
  • $ -(a\!-\!b) = b\!-\!a$
  • $ -\sum_{j = 1}^n a_j = \sum_{j = 1}^n (-a_j)$
  • $ -\int_a^b f(x)\,dx = \int_b^a f(x)\,dx$
Exactly similar properties (except the last) are valid in every ring. The fourth property implies the

Corollary. If one changes the sign of one factor of a ring product, then the sign of the whole product changes.



"opposite number" is owned by pahio.
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See Also: ring, opposite polynomial, condition of orthogonality, automorphism, product of negative numbers, plus sign

Other names:  negative [as a noun]
Keywords:  addition

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Cross-references: product, ring product, factor, implies, ring, similar, properties
There are 10 references to this entry.

This is version 5 of opposite number, born on 2005-02-18, modified 2006-12-08.
Object id is 6775, canonical name is OppositeNumber.
Accessed 7331 times total.

Classification:
AMS MSC: 12D99 (Field theory and polynomials :: Real and complex fields :: Miscellaneous)
 97D99 (Mathematics education :: Education and instruction in mathematics :: Miscellaneous)
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